Background
In this example, we will show how to use the Emulator to prepare ordered ground states in Rydberg systems. The example is based on experimental works in 1D system, and 2D system.
Due to the strong Rydberg interactions, only one Rydberg excitation is allowed within the so-called blocakde radius Blockade. With positive detunings, more Rydberg excitations, however, are favored to lower the energies of the ground states. With the interplay of these two mechanisms, different ordered states are supported depending on the strength of blockade radius and the detuning. For example, in 1D there exists the $Z_N$ ordered states, and in 2D there are the checkboard phase, star phase, and even pure quantum phase (straited phase).
The adibatic process works as follows: One starts from all the atoms being in the ground state, which is also the ground states of many-body Hamiltonian with large negative detuning. Then the Rabi freuency $\Omega$ is turned on, and the detuning strength is ramped up from large negative value to postive values. If such process is slow enough, the quantum state the system stays close to the ground state of the instantaneous Hamiltonian. At the end of the process, one arrives a target Hamiltonian, and correspondingly, the prepared state is approximately the ground state for the final Hamiltonian.
We start by importing required libraries
using EaRyd
using CairoMakie
using EaRydPlotsOne dimensional case
We first specify the adiabatic pulse sequence for Rabi frequency by using the built-in waveform function piecewise_linear
total_time = 3
Ω_max = 2π * 4
Ω = piecewise_linear(clocks=[0.0, 0.1, 2.1, 2.2, total_time], values=[0.0, Ω_max, Ω_max, 0, 0])
draw(Ω)
#Δ1 = piecewise_linear(clocks=[0.0, 0.6, 2.1, 3], values=[-10*2π, -10*2π, 10*2π, 10* 2π])
The detuning sequence could also be created in a similar way
U1 = -2π * 10
U2 = 2π * 10
Δ = piecewise_linear(clocks=[0.0, 0.6, 2.1, total_time], values=[U1, U1, U2, U2])
draw(Δ)
We then generate the positions of a 1D atomic chain by using the function generate_sites
nsites = 9
atoms = generate_sites(ChainLattice(), nsites, scale=5.72)Note that we have specify the nearest-neighbor atoms being seperated by 5.72 $\mu m$ in order to prepare a $Z_2$ ordered state. With the waveforms and atomic coordinate specified, the time-dependent Hamiltonian can be simply generated by the following
h = rydberg_h(atoms; C=2 * pi * 858386, Δ, Ω)Hamiltonian
Term 1
∑(n=1:9) Waveform(_, 3.0)(t)/2 σ^x
Term 2
-∑(n=1:9) Waveform(_, 3.0)(t) n
Term 3
∑(n=1:9) 5.39e6/|r_i - r_j|^6 n_i n_jWe then specify all atoms to be in ground state initially, and set up the emulation problem by choosing the ODE solver
reg = zero_state(9)
prob = ODEEvolution(reg, total_time, h; dt=1e-5, adaptive=false)ODEEvolution{Float64}:
reg: ArrayReg{2, ComplexF64, Matrix{ComplexF64}}
reg storage: 8.000 KiB
timespan: (0, 3)
equation:
Schrödinger Equation:
Storage Size: 8.055 KiB
State Storage: Vector{ComplexF64}
Hamiltonian Storage: EaRydCore.ConstTermCache{Tuple{EaRydCore.var"#94#127", Waveform{EaRydWaveforms.PiecewiseLinear{Float64, Interpolations.Extrapolation{Float64, 1, Interpolations.GriddedInterpolation{Float64, 1, Float64, Interpolations.Gridded{Interpolations.Linear{Interpolations.Throw{Interpolations.OnGrid}}}, Tuple{Vector{Float64}}}, Interpolations.Gridded{Interpolations.Linear{Interpolations.Throw{Interpolations.OnGrid}}}, Interpolations.Throw{Nothing}}}, Float64}, typeof(EaRydCore._const_param_)}, Tuple{LinearAlgebra.Transpose{Float64, SparseArrays.SparseMatrixCSC{Float64, Int32}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}}}
Hamiltonian
Term 1
∑(n=1:9) Waveform(_, 3.0)(t)/2 σ^x
Term 2
-∑(n=1:9) Waveform(_, 3.0)(t) n
Term 3
∑(n=1:9) 5.39e6/|r_i - r_j|^6 n_i n_j
options:
algo: Vern8(true)
progress: false
progress_steps: 5
progress_name: "ODE"
reltol: 1.0e-8
abstol: 1.0e-8
normalize_steps: 5
normalize_finally: trueWe measure the Rydberg density for each site and time step
densities = []
for info in prob
push!(densities, [expect(put(nsites, i=>Op.n), info.reg) for i in 1:nsites])
end
D = hcat(densities...)